We will now present two explicit realizations of the mechanism we just described. In order to study how FI-terms arise in the reducedN = 1 theory we only need to consider the the quaternionic manifold and its isometries. We will consider a model with a single hypermultiplet with the kinetic terms charaterized by the quaternionic manifold MQ= SO(4,1) SO(4) ∼ = Sp(1,1) Sp(1) Sp(1). (6.9.5)
We define the scalars in the hypermultiplets{h, b1, b2, b3}so that the correspond-
ing quaternionic vielbein and SU(2) connection are given by
fiA= √1 2(dh 2+ ie −hσxdb x), ωx=−12e −hdb x. (6.9.6)
Therefore, the kinetic terms of the hyperscalars read:
T =−∂µh ∂µh−e−2h∂µ~b ∂µ~b, ~b= (b1, b2, b3). (6.9.7)
We will study the effect of gauging the isometries of the quaternionic manifold characterized by the following killing vectors:
k1 = 2b1 ∂ ∂b2 −2b2 ∂ ∂b1 , k2 = 4b3 ∂ ∂h + 4b1b3 ∂ ∂b1 + 4b2b3 ∂ ∂b2 + 2[b23−b21−b22+ 1−e2h] ∂ ∂b3 . (6.9.8) These two killing vectors induce a rotation of the complex structures, and there- fore the corresponding compensator is non-trivial (6.5.9):
~rk1 = 0 0 1 , ~rk2 = 2 −b2 b1 −eh . (6.9.9)
The moment maps can be computed using the equations (6.5.10),
~ Pk1 = e−hb2 −e−hb 1 1 , P~k2 = −2b2−2b1b3e−h 2b1−2b2b3e−h −e−h[b2 3−b21−b22+ 1]−eh . (6.9.10)
6.9. N = 1 FI terms from N = 2 supergravity.
Field independent compensator.
From (6.9.9), we can see that the gauging of the killing vector η1k1 leads to
the situation described in the previous paragraph. We can define a reduction of supersymmetry consistent with the condition k1= 0, which impliesb1=b2= 0.
The kinetic terms of the reduced theory read:
T =−∂µh ∂µh−e−2h∂µb3∂µb3, (6.9.11)
and definingS= eh+ ib3 they can be derived from the K¨ahler potential
K(S,S¯) =−log(S+ ¯S). (6.9.12) Actually, the scalar manifold of the reducedN = 1 supergravity theory is SU(1U(1),1), which describes the axio-dilaton system that we studied in previous chapter. Ac- cording to the identifications (6.8.7b) the reducedN = 1 theory has a vanishing superpotential, and the compensator~rk1 leads to a constant contribution in the
moment map of the reduced theory, a FI-term:
P|N=1= 2Pk31|k1=0= 2η1. (6.9.13)
Note that, as the killing vectork1vanishes identically after the truncation, there
is no gauged symmetry in the reduced action. However, if we gauge a symmetry
k=η1k1+k3, wherek3 acts non-trivially on handb3, we can end in a N = 1
supergravity action with an abelian gauge symmetry and a constant FI-term.
Field dependent compensator.
Consider now the gauging ofη2k2. In the reduced theory defined byb1=b2= 0,
k2 still acts non trivially onhandb3:
k2|N=1= 4η2b3 ∂ ∂h+ 2η2[b 2 3+ 1−e 2h] ∂ ∂b3 , (6.9.14)
which can written in terms ofS as
k2|N=1= i2η2(1−S2)
∂
∂S. (6.9.15)
Thus the killing vector k2 represents the same isometry we used in sections 5.4
and 5.4.2 to obtain constant FI-terms from field dependent moment maps (5.4.6) after the consistent truncation of the axio-dilaton. Actually, after the truncation N = 2→ N = 1 the moment map of this killing vector reduces to
P|N=1= 2Pk32|b1=b2=0=−2η2[e −h(b2 3+ 1) + e h] = −4η2 1 +|S|2 (S+ ¯S). (6.9.16)
N = 2 supergravity and effective Fayet-Iliopoulos terms.
As we discussed in the previous chapter, when the fieldS is truncated atS= 1, the fixed point ofk2|N=1, this moment map also becomes an effective FI-term:
P(S= 0)|N=1=−4η2. (6.9.17)
Note that, although the compensator associated tok2 was not a constant in the
original N = 2 theory, it has led to a constant FI-term in the reduced theory. Thus, the mechanisms to generate FI-terms from consistent reductions ofN = 2 theories are more general that the one discussed in [73].
CHAPTER
7
Supersymmetric cosmic strings in
N
= 2
supergravity.
7.1
Introduction.
The present chapter is dedicated to the embedding of local supersymmetric cosmic string solutions inN = 2 supergravity. The first known example of such string solutions was constructed in [73]. The authors of this work considered a model with the minimal matter content needed to obtain a half-BPS cosmic string solution inN = 2 supergravity action: one hypermultiplet and one vector multiplet.
As in N = 1 supergravity, (see section 2.5), the supersymmetric cosmic strings are solutions of the BPS equations, which are obtained imposing the field configuration to preserve half of the supersymmetries of the system. In [73], only field configurations compatible with a consistent truncation to N = 1 were considered, i.e. the cosmic strings are also valid solutions for a N = 1 supergravity model. Although these string configurations involve only the fields surviving the truncation, supersymmetry is not spontaneously broken to N = 1 supergravity, and the full N = 2 supersymmetry is preserved in the vacuum far away from the string. From a N = 2 point of view, the use of an
Supersymmetric cosmic strings in N = 2 supergravity.
ansatz related to a consistent truncation to N = 1 supergravity in [73] is not required but is useful to simplify the calculation, as the N = 2 BPS equations are in general much more difficult to solve than those of N = 1 theories. The cosmic string solution found in [73] can be seen as an embedding of aN = 1 D-term string inN = 2 supergravity. This is consistent with the discussion in [61] (reviewed in chapter 2), where it is argued that the only N = 1 models admitting supersymmetric vortex solutions involve aD-term potential endowed with a constant Fayet-Iliopoulos term.
The main purpose of the work presented in this chapter is to enlarge the type of N = 2 supergravity theories that can generate constant FI terms in N = 1 supergravity. The special geometry used in [73] is a very particular case, it corresponds to the so-called minimal special geometry, which is based on a quadratic prepotential. Here we realize the construction of [73] with a special geometry based on acubic prepotential.
For simplicity we will study a model with abelian gauging of isometries, and thus we will not be able to avoid the presence of hypermultiplets. Indeed, the hy- permultiplet is required in order to provide the scalar acting as a Higgs field since, for Abelian gauging, supersymmetry forbids the scalars of vector multiplets to be charged under gauge transformations. As we discussed in the previous chapter, in the presence of hypermultiplets it is not possible to have constant FI-term in a N = 2 supergravity model. Therefore, the FI-terms of the correspond- ing reducedN = 1 theories are generated from field dependent moment maps in the motherN = 2 theory, using mechanisms described in the previous chapter. The field content of the model that we are going to study consists of one hypermultiplet and two vector multiplets, and the couplings are characterized by the following choice of quaternionicMQand special K¨ahler manifoldsMSK:
MQ= SO(4,1) SO(4) MSK=ST[2, n] = SU(1,1) U(1) × SO(2, n) SO(2)×SO(n). (7.1.1) In order to define the gauge couplings the we will use the so called Calabi- Vesentini symplectic section [139, 152], well-known from different compactifica- tions of string theory [145, 146]. An interesting feature of this section is that it can be used to construct models that exhibit partial breaking of supersymmetry N = 2 → N = 1 [153]. In this model we gauge a U(1) subgroup of the R-symmetry that rotates the complex structures of the quaternionic manifold. The corresponding compensator reduces to a constant after the truncation, which acts as an effective FI -term (see section 6.9). The truncated theory of this model contains an axiodilaton fieldS=a−ieρ, which appears in the gauge
kinetic function that defines the kinetic term of the vector field isf =iS. We shall see that theN = 2 BPS equations imply that, in the background of the string, the axiodilaton must be an arbitrary constant. Once the winding number